line A B, and therefore the wheel o, to revolve uniformly. This is of course true whether the wheels move in the same or in contrary directions to those indicated by the arrow-heads in fig. 149. In order to prevent the locking of the teeth, it is usual to make A E less than EA, by of the pitch AA, ; and to cut the space A B' deeper than the perpendicular length of the tooth AC in such a manner that the distance from c to the centre is less than the distance from B' to the same centre by of the pitch ▲ A1; if, however, the workmanship is very good, the differences can in both cases be made smaller.

*

ΤΟ

The rule for determining the length of the teeth commonly adopted by millwrights is to make the length of the tooth beyond the pitch circle (i.e. a c or a c′) equal to of the pitch. This rule is, however, a very bad one; the following, though not perhaps the best, is very much better. Suppose o to be the driver, and suppose a pair of teeth to be in contact on the line of centres, the face of the next tooth should be so long that its extreme point c2 should just be on the circumference of the generating circle AX, as shown in the figure; the length of the tooth of the follower is determined by a similar rule; the extreme point of the following tooth c, should (under the same circumstances) be on the circumference of the generating circle A x 0. The reason of this rule is as follows: may be considered that when the wheels are in motion that pair will bear the whole or nearly the whole strain which at any instant will be the next to go out of contact ; so that, the above construction being employed, the one pair of teeth is just going out of contact when the next pair comes to the line of centres, and consequently the working strain is not thrown upon any pair of teeth until

it

* Willis's Principles of Mechanism, p. 98. The rule which follows is given both by Mr. Moseley, Mechanical Principles, p. 267, and by Gen. Morin, Aide-Mémoire, p. 280.

it comes to the line of centres; but it appears that practically the friction between a pair of teeth is very much more destructive when they are in contact before the line of centres than when in contact behind the line of centres; by following, therefore, the rule above given, the friction between any pair of teeth is diminished. (Compare Ex. 566.)

In practice the teeth of a wheel are all cut from a pattern; in constructing a pattern the epicycloidal curve may be drawn by the actual rolling of a circle of the proper size; or an approximation may be obtained by means of circular arcs. Rules proper for this purpose will be found in Mr. Willis's Treatise above referred to.

Ex. 544.-To determine the radius of the pitch circle of a wheel which shall contain n teeth of given pitch a.

α

Ans. r

Ex. 545.-If a wheel of m teeth drives another of n teeth; then if the driver make p revolutions per minute the follower will make mp revolutions per minute.

n

Ex. 546.-There are three parallel axes, A, B, C ; ▲ makes p revolutions per minute, it carries a wheel of m1 teeth which works with a wheel of n1 teeth on B; B also carries another wheel of m2 teeth which works with a

wheel of n2 teeth on c; show that c makes minute.*

Ρ

revolutions per

Ex. 547.-A winding engine is worked in the following manner: a steam engine causes a crank to make 30 revolutions per minute; the axle of the crank has on it a wheel containing 36 teeth, which works with a wheel containing 108 teeth, the latter wheel is on the same axle as the drum, which is 5 ft. in radius; determine the number of feet per minute described by the load. Ans. 314 ft.

*The above arrangement is to be found in most cranes; if the student is not acquainted with the arrangement of a train of wheels he will do well to examine a good crane, such as is to be seen at most railway stations : the train of wheels in a clock is also a good example, but cannot commonly be studied without taking the clock to pieces.

110. The Hunting Cog.-If wheels have to do heavy work, and the precise proportion between the velocities is not of great importance, an additional tooth-called a hunting cog-is introduced into one of the wheels, so that the same pair of teeth may seldom work together; by this means they are kept from wearing unequally. For instance, if in the last Example we denote the teeth of the driver by the successive numbers 1, 2, 3,

36, and the teeth of the follower by the successive numbers 1, 2, 3, .. 108; then in every revolution 1 will work with 1, 37, and 73; 2 will work with 2, 38, and 74; and 36 will work with 36, 72, and 108. If now we introduce a hunting cog into the driving-wheel, so that it contains 37 teeth, then on the first revolution 1 will work with 1, 38, and 75; in the next revolution with 4, 41, and 78; in the third with 7, 44, and 81, and not until the 38th revolution will it work with 1 again.

Ex. 548.—If in the last Example a hunting cog' were introduced into the driver so that it contains 37 teeth, determine the number of feet per minute the load will now travel. Ans. 323 ft.

Ex. 549.-If in Ex. 546 there are k+1 axles and the drivers contain m teeth, and the followers contain n teeth a-piece, show that the number of

revolutions made by the last axle will be p (m)*

k

Ex. 550.—If in the last Example it is required to multiply the number of revolutions 200 times, how many axles must we use-(1) if we take m=2n; (2) if we take m=4n; (3) if we take m= =6n, and determine the number of teeth employed, in each case using the nearest whole numbers?

Ex. 551.-If each driver has m teeth, and each follower n teeth, and if M is the total number of teeth in the train, and if the last axle makes q revolutions while the first axle makes one revolution, show that

M

m + n

(C)-5

* Ex. 552.-In the last Example show that for given values of м and n we shall obtain the greatest value of q by making m=3·59. n nearly.* whence the result stated.]

[It is easily shown that loge (m) = 1 +

n

m

Ex. 553.—In the case of a pair of wheels with epicycloidal teeth show that the space through which the surfaces of each pair of teeth slide one upon the other while in contact and after passing the line of centre is ap

2πη

proximately represented by the formula 2(+)

or

n1

(+)

where r and r1 are the radii of the driver and follower respectively, and n and n, the number of teeth in those wheels respectively.

[The motion of one tooth on the other is partly a sliding and partly a rolling motion. Now, if we refer to fig. 150 it is evident that the pair of teeth just going out of contact touch at c2; it is also evident that the two points A, and A'2 were in contact at A, so that the space through which the surfaces have slidden over each other is A2 A2, which is very nearly equal to the sum of the versed signs of the arcs A A, and ▲▲′2, i.e. to r vers n

2π

+1 vers ; whence the value assigned in the question.]

n1

2π

Ex. 554.-A weight P balances a weight q under the following circumstances; P is tied to a rope which is wrapped round an axle whose radius is p; is tied to a rope which is wrapped round an axle whose radius is q; to the former is attached a concentric rough wheel, whose radius is r, to the latter in like manner a concentric rough wheel, whose radius is r1; these two wheels

FIG. 151.

Q

* It would appear from this that the best proportion between the number of teeth in driver and follower for multiplying velocity is 1:4. This result is due to Dr. Young, Lectures, vol. ii. p. 56. Mr. Willis remarks that the rule is not of much practical value, Principles, p. 218.

is evident that the rough wheels act on each other by means of a mutual action through the point a.]

Ex. 555.—In the last Example if we suppose the separate wheel and axles to turn round axes whose radii are p and p1 respectively and the limiting angles of resistance between them and their bearings to be and $1, show that when P is on the point of overcoming Q we have the following relation (neglecting the rigidity of cords, and the weights of the wheel and axles)

P(p−p sin p) (~1+P1 sin $1)=Q(Q+P1 sin Q1) (r+p sin ọ)

Ex. 556.—If in the last Example, besides the frictions on the axes, we take into account the weights w and w1 of the wheel and axles, determine the relation between P and Q.

Ex. 557.—If in the last Example we neglect powers and products of p sín o p sin ☀ P1 sin P1, P1 sin Q1, show that the number of units of

work that must be done in order to raise a weight of a lbs. through a space of 8 ft. is given by the formula

Ex. 558.—In the last Example if we suppose the rough wheels to be replaced by a pair of toothed wheels whose pitch circles have the same radii as the wheels; then if the wheel o contains n teeth, and the wheel o1 contains n, teeth, show that when Q is raised through a space of s ft. the work lost by the friction of the teeth is approximately represented by the formula

μας

π

π

+ where μ is the coefficient of friction between the teeth.

n n

2π Պլ

[If the wheel o1A revolves through an angle the space

the surfaces of the driving and driven teeth slide is

and therefore, supposing R, the mutual pressure, to continue constant during the contact of the teeth, the number of units of work expended 2πηι π on friction equals με n1

π

(+). Now, approximately,

n

Rr1 =Qq, and therefore the work expended on one pair of teeth equals 2 is the space through which Q is raised during

П

n

:;);

If P instead of being a weight were a force acting vertically upward it is easily shown that the third term of this equation is